Baby Universes, Holography, and the Swampland
BBaby Universes, Holography, and the Swampland
Jacob McNamara and Cumrun Vafa
Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA
Abstract
On the basis of a number of Swampland conditions, we argue that the Hilbert spaceof baby universe states must be one-dimensional in a consistent theory of quantumgravity. This scenario may be interpreted as a type of “Gauss’s law for entropy”in quantum gravity, and provides a clean synthesis of the tension between Euclideanwormholes and a standard interpretation of the holographic dictionary, with no needfor an ensemble. Our perspective relies crucially on the recently-proposed potential forquantum-mechanical gauge redundancies between states of the universe with differenttopologies. We further comment on the possible exceptions in d ≤ d = 2 (such as JT gravity and possible cousins in d = 3), which we argue areincomplete physical theories that should be viewed as branes in a higher dimensionaltheory of quantum gravity for which an ensemble plays no role. a r X i v : . [ h e p - t h ] M a y ontents α -Parameters 4 α -Parameters and Coupling Constants . . . . . . . . . . . . . . . . . . . . . 52.2 Potential Loss of Unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Gauge Redundancy in the Baby Universe Hilbert Space . . . . . . . . . . . . 8 − − α -Parameters . . . . . . . . . . . . . . . 113.3 No Global Symmetries as a Consequence . . . . . . . . . . . . . . . . . . . . 12 One of the basic lessons of string theory is that, in quantum gravity, anything that can bedynamical must be dynamical. In particular, every coupling constant in string theory isthe asymptotic value of a dynamical field, and every symmetry is coupled to a dynamicalgauge field. This is a version of background independence, since otherwise we would have atheory of quantum gravity that depends on a choice of fixed background parameters. Thecondition that quantum gravity must have no free parameters has been codified as one of themost basic swampland conditions (for a review of the Swampland Program, see [1, 2]), andhas been proposed to hold in all consistent theories of quantum gravity in d ≥ d ≥ d ≤
3. Indeed there are obvious counterexamples in d = 2 to the lack of freeparameters. In particular, any coupling constant of the worldsheet CFT (such as the radii of1nternal geometries) is a free parameter of the two-dimensional theory of quantum gravity onthe string worldsheet. Related to this fact, note that in d = 2, massless scalar fields do notget a vev, and do not lead to different superselection sectors in infinite spatial volume. Thus,we cannot hope to realize a parameter of the theory as the vacuum expectation value of amassless dynamical field. More generally, many swampland conditions are violated in d = 2,such as the absence of global symmetries and the triviality of cobordism. The case d = 3may admit similar exceptions because gravity has no propagating degrees of freedom, andthere might exist topological theories of gravity (such as gravitational Chern-Simons theory)which do not follow the usual swampland conditions of higher dimensional quantum theoriesof gravity. To cover these potential exceptions in d = 3, we can replace the restriction of d ≥ d > d = 2 holography in the context of JT gravity [3]. Inparticular, if one considers a collection of k one-dimensional quantum systems, dependingon some free parameters averaged over an ensemble, one obtains a holographic quantumgravity dual in d = 2, which entails summing over bulk geometries possibly connecting the k boundaries. Moreover, changing the measure for this ensemble will change the parame-ters of the gravitational theory, and so the d = 2 bulk gravity would inherit free unfixedparameters. It is tempting to ask whether this lesson learned in d = 2 can be exported tohigher dimensions, and should lead to a departure from the standard AdS/CFT dictionaryin higher dimensions, forcing us to contend with an ensemble of dual boundary theories.This ensemble average may be interpreted as the result of an old, well known source offree parameters in quantum gravity [3], namely the so-called α -parameters of Coleman [4],studied further in [5, 6] (for a review and other connections to the Swampland Program, see[7]). In the Euclidean path integral, spacetime wormholes can be interpreted as calculatingamplitudes to produce or absorb baby universes. These processes pose a threat to unitarityof the quantum system, in the form of potential information loss [4, 5] or non-factorizationof correlation functions in a holographic dual [8]. The proposed resolution is to suppose thebaby universes are in a specific α -eigenstate (in which case there are no issues with unitarityand factorization) at the cost of introducing α as free parameters of the theory, which are notthe expectation value of any dynamical fields. Thus, we see an immediate tension betweenthe Euclidean path integral and the expectation from the Swampland Program that quantumgravity should have no free parameters. Of course, a massive parameter could be realized as the vev of a massive field. α -parameters andthe baby universe Hilbert space in a two-dimensional toy model [9]. While they do stillfind nontrivial α -parameters, a key ingredient in their story is that the Euclidean path in-tegral implies an enormous redundancy in the naive baby universe Hilbert space, leadingto an enormous reduction in the number of degrees of freedom. Essentially, cobordismsconnecting baby universe states with different topologies act as a generalized form of gaugetransformation, which force the gauge-invariant wavefunctions to include specific superpo-sitions of baby universes with different topologies, an idea which has also been consideredin [10]. Chief among the gauge invariant wavefunctions is the Hartle-Hawking wavefunction[11], defined by the Euclidean path integral with no initial boundary, but there are manyother gauge-invariant states in their toy model, leading to many different α -eigenstates.What, then, is the lesson for quantum gravity in dimension d >
3? There seems tobe a paradox: the general considerations leading to α -parameters make no reference tothe dimension, but the Swampland Program heavily suggests that there should be no freeparameters in a theory of quantum theory of Einstein gravity in d >
3. However, there is aclean resolution, which is to suppose that the gauge redundancies described in [9, 10] are sostrong in d > α -parameters, and so there would beno tension between the Euclidean path integral and our understanding of quantum gravityin d >
3. In addition, as pointed out in [9], in this case there would be no need to discussmodifications of the standard holographic dictionary involving ensembles.In this note, we argue that this indeed must happen in a consistent theory of quantumgravity in d >
3. That is, we propose the following hypothesis as a swampland condition.
Baby Universe Hypothesis.
Let H BU be the Hilbert space of baby universes in a unitarytheory of quantum gravity in d > spacetime dimensions. Then we have dim H BU = 1 . Put differently, we propose that the only gauge invariant state of baby universes is the Hartle-Hawking wavefunction. As a swampland condition, this hypothesis should place enormousconstraints on which effective field theories admit a realization in quantum gravity. Indeed,the Baby Universe Hypothesis will be massively violated with a generic choice of matter andinteractions, and looks quite miraculous from the perspective of effective field theory.As a consequence, we conclude that the ensemble interpretation of holography is verymuch a feature of d = 2 and potentially d = 3 theories of gravity, for which many swamplandprinciples do not apply. Further, we explain why the presence of an ensemble in d = 23including the case of JT gravity) is actually already anticipated by the standard (non-ensemble) holography of higher dimensional theories, by viewing a d = 2 spacetime as the‘t Hoof t worldsheet associated to the standard holographic duality of a higher dimensionalgauge theory. That there is no known analog of higher dimensional ( d >
3) branes that canserve as a large N perturbative expansion of some QFT system is beautifully compatiblewith our hypothesis that we do not expect an ensemble interpretation of holography to existfor d >
3. We argue that any such low dimensional exceptions, including JT gravity, areincomplete physical theories: they should not be viewed as standalone quantum gravitationalsystems, but rather as worldvolume theories of branes in higher dimensional theories ofquantum gravity that do not enjoy any such exceptions. In particular, the large ambiguityin choosing an ensemble is related to the ambiguity in the choice of which observable tomeasure in the higher-dimensional quantum system.This note is organized as follows. In Section 2, we review Coleman’s argument [4] for α -parameters and their implications for unitarity, as well as the potential for an enormous gaugeredundancy [9, 10] in the baby universe Hilbert space. In Section 3, we argue for the BabyUniverse Hypothesis on the basis of a number of swampland conditions, as well as derive theabsence of global symmetries as a consequence of the Baby Universe Hypothesis. In Section4, we describe the implications of the Baby Universe Hypothesis for holography, both inthe context of AdS/CFT and more broadly as an interpretation of the general holographicprinciple as “Gauss’s law for entropy.” We also discuss why the ensemble interpretation forholography in the case of d = 2 (and potentially 3) is natural, and why we do not expect thisexception to persist in higher dimensions. Finally, in Section 5, we conclude our discussion. α -Parameters In this section, we review the arguments [4, 5, 6] that Euclidean wormholes and the pro-duction of baby universes can lead to an ensemble of quantum systems, labeled by so-called α -parameters, as well as the potential loss and restoration of unitarity. Further, we reviewthe picture outlined in [9, 10] of a potentially enormous gauge redundancy in quantum grav-ity relating states with different topologies, and its role in defining the baby universe Hilbert A similar perspective on the presence of an ensemble and its relationship with the computation of Wilsonloop observables has appeared in [12] in the context of c = 1 Liouville theory. From this perspective, we believe that there is no exception to our hypothesis for a complete quantumgravitational system in any dimension. In particular, we expect that all quantum theories of gravity whicharise by compactification of string theory down to d = 2 and 3 dimensions pose no exceptions to the conditionof no free parameters. α -Parameters and Coupling Constants In [4], Coleman examines the fact that the Euclidean path integral naturally includes config-urations that can be interpreted as tunneling amplitudes for a small piece of the universe todetach as a disconnected baby universe, only to reattach itself at another point in spacetime.Coleman outlines how attempting to integrate out the effects of these Euclidean wormholesleads to an ensemble of bulk theories, labeled by different α -parameters, which label states ofthe baby universes in which no quantum coherence is lost. These α -parameters correspondto coupling constants in the action of our effective field theory. In this section, we reviewColeman’s argument, highlighting the features that will play a key role for us below. Baby Universe 𝑡 " A baby universe detaches and reattaches in Euclidean time.The basic argument presented in [4] is as follows. Suppose that there are many speciesof baby universe, labeled by a discrete variable i . We may define baby universe creation andannihilation operators a † i , a i , which satisfy the standard bosonic commutation relations. Interms of these operators, the effective action may be written as S = S + (cid:88) i ( a † i ∗ + a i ) (cid:90) L i , where i ∗ labels the CPT conjugate of the baby universe state i , and where the absorption ofbaby universe i inserts the local operator L i in the effective field theory. Notably, we have For mathematicians, this argument may be summarized by saying that the baby universe Hilbert spaceis naturally a commutative Frobenius algebra under disjoint union, together with the structure theorem forsuch algebras. a † i ∗ , a i outside of the integral over spacetime, as they are independentof position in spacetime (put differently, the baby universes carry zero momentum).Now, we define A i = a † i ∗ + a i . The operators A i all mutually commute, which allows us to define mutual eigenstates | α (cid:105) ,where we have A i | α (cid:105) = α i | α (cid:105) . In addition, the operators A i all commute with the Hamiltonian, and so the different α -eigenstates define different superselection sectors of the effective field theory. In a fixed α -eigenstate, the effective action takes the form S = S + (cid:88) i α i (cid:90) L i , and so we see that the α -parameters serve as coupling constants for different terms in theeffective action. Suppose we begin in a state of the universe which is a tensor product of some fixed babyuniverse state and a state of the effective field theory, that is, suppose | ψ (cid:105) = | ψ BU (cid:105) ⊗ | ψ EFT (cid:105) . Generically, under time evolution, such a state will evolve into an entangled state, | ψ (cid:48) (cid:105) = (cid:88) i c i | ψ i BU (cid:105) ⊗ | ψ i EFT (cid:105) . If we trace out the baby universe state, we see that from the perspective of effective fieldtheory, a pure state has evolved into a mixed state | ψ EFT (cid:105) (cid:104) ψ EFT | (cid:32) (cid:88) i | c i | | ψ i EFT (cid:105) (cid:104) ψ i EFT | , and so information has been lost. Put differently, the state of the universe is becomingentangled with the state of the baby universes, which looks to an observer in the effective6eld theory like a loss of information [13].Though in a strict sense information is being lost in this process, it has been argued thatthe existence of α -eigenstates ensures that nothing catastrophic arises from this informationloss. First of all, if | ψ BU (cid:105) = | α (cid:105) , is initially an α -eigenstate, then since the Hamiltonian only interacts with the baby universeHilbert space through the operators A i , the baby universes will remain in an α -eigenstatefor all time, and so there will be no transfer of information to the baby universes. Thus,each different choice of α -parameters defines a well-defined and unitary effective field theory,which may be thought of as different superselection sectors. These different effective fieldtheories differ only in the values of their coupling constants.What are we supposed to think if the baby universes are in a superposition of the different α -eigenstates? In this case, information is indeed lost. However, though we lose track of therelative phases of the different α -eigenstates encoded in the initial state | ψ BU (cid:105) , we could notdetect this information to begin with. Since we can only interact with the baby universeHilbert space through the mutually commuting operators A i , we should think of a state | ψ BU (cid:105) as only encoding a classical mixture of states with different α . Put differently, thesystem describes a classical ensemble of different effective field theories, with a probabilitydistribution on α corresponding to our ignorance of the exact values of coupling constants.Thus, while the results of quantum gravitational experiments (such as black hole formationand decay) cannot be predicted exactly, this will be interpreted as a lack of precise knowledgeof coupling constants, as opposed to a contradiction with unitary time evolution.Though this argument holds up to scrutiny, already we see that there is an another,much cleaner scenario for preserving unitarity. This is to suppose that for some mysteriousreason, the Hilbert space of baby universes must be one-dimensional! In this case, the babyuniverses would have no capacity for storing information, and so even though informationtries to lose itself in the state of the baby universes, the vanishing entropy of closed universesforces information to be preserved in the effective field theory! At this stage, this hypothesismerely trades discomfort at a formal, yet undetectable, loss of information in a theory with α -parameters for a perhaps much greater discomfort at the idea that the baby universeHilbert space is somehow one-dimensional. However, as we argue below, this is the naturalconclusion from the perspective of the Swampland Program, and thus our discomfort can beinterpreted as a relic of trying to apply the generic expectations of effective field theory tothe non-generic case of quantum gravity! 7 .3 Gauge Redundancy in the Baby Universe Hilbert Space How could it be that the Hilbert space H BU of baby universe states is one-dimensional? Anaive count of degrees of freedom would suggest a much larger dimension, since from theperspective of effective field theory, there are fluctuating degrees of freedom attached toeach point in space, and moreover there seem to be degrees of freedom corresponding tothe topology of the baby universes. This is the basic tension between effective field theoryand the holographic principle, in that there must be some mechanism in non-perturbativequantum gravity that greatly reduces the number of physically distinct configurations, inorder to satisfy an area law for the entropy.Such a mechanism has recently been proposed and studied in [9, 10], which is to realizethat there are gauge redundancies which can identify wavefunctions of the universe withdifferent topologies. One place this can be seen clearly is in the case of the two sided AdS-Schwarzschild black hole, which is dual to the thermofield double state in AdS/CFT [14], aquantum superposition of states with disconnected spatial topology. As pointed out in [10],the fact that the two-sided black hole is not orthogonal to disconnected states may be seenfrom the Euclidean path integral, which includes configurations that change the topology ofthe spatial slice, leading to a nonzero inner product between states with different topology. |𝜓 ⟩⟨𝜓 & | A contribution to the inner product (cid:104) ψ | ψ (cid:105) .In the case of the baby universe Hilbert space, as has been recently discussed in [9], thesegauge redundancies may be computed as follows, in close analogy with the reconstruction ofthe Hilbert space from Euclidean correlators in axiomatic quantum field theory. Heuristically,we start with a larger vector space (cid:101) H BU of putative states of baby universes, given by finitequantum superpositions of classical states of the baby universes (in the context of AdS/CFT,89] describes these states by fixing the asymptotic boundary conditions in the Euclidean past).The Eucludean path integral over geometries and topologies defines an inner product on thisvector space, which is positive semi-definite by reflection positivity. Taking the Hilbert spacecompletion yields the Hilbert space H BU of baby universes.If the inner product computed on (cid:101) H BU is not positive definite, taking the completionincludes a quotient by the space of null states, namely baby-universe wavefunctions of zeronorm. Thus, wavefunctions differing by a null state yield the same physical state in H BU ,which may be viewed as a form of gauge redundancy. An important difference from thecase of an ordinary gauge theory is that the physical Hilbert space cannot be viewed asarbitrary wavefunctions of some gauge-fixed variables, at least not in any obvious way. Thisis related to the fact that while the constraint equation in ordinary gauge theory is a first-order differential equation, whose solutions correspond to wavefunctions on the quotientspace, the Wheeler-de Witt equation is second order, and induces quantum redundanciesbetween wavefunctions of a more complicated nature [15]. In this section, we argue for the Baby Universe Hypothesis on the basis of the SwamplandProgram. As a result, we conclude that in d >
3, the gauge redundancies described inthe previous section are so great that they cut the baby universe Hilbert space down to asingle state. First, we review how the condition of no free parameters should be naturallyviewed as part of the statement that there are no generalized global symmetries in quantumgravity. We then argue that nontrivial α -parameters would violate this condition. Finally, weprovide further evidence that the Baby Universe Hypothesis fits naturally into the growingweb of swampland conjectures, by showing that it implies the abscence of ordinary globalsymmetries in quantum gravity. ( − -Form Symmetries Our experience with string theory suggests that there cannot be any free parameters inquantum gravity in dimension d >
3. What we mean by this is the following. Suppose wehave a low energy effective theory of quantum gravity in d > − λ in the Lagrangian is naturally associated with a d -formoperator L λ , such that the effective action takes the form S ( λ ) = S + λ (cid:90) L Λ . Since L λ is a top form, it is naturally closed, and so we may define a 0-form conserved current J λ = (cid:63) L λ , which we may view as the Noether current for the associated ( − d -volume operators that implement the “symmetry” cannot be placed back-to-back, and sothere is no need for a group law. Thus, just as p -form symmetries for p ≥ − If we couple L λ to a dynamical field φ rather than a free parameter λ , in that we have S = S + (cid:90) φ L λ , this corresponds to gauging the ( − This situation should be compared to the homotopy groups of a space X . While π k ( X ) is an abeliangroup for k ≥ π ( X ) is just a group, and π ( X ) is just a set. φ (which may be identified with λ ) define superselection sectors for d ≥ p -form symmetry in quantum gravity must be gauged for the case p = −
1. In particular, this includes both continuous and discrete parameters, just as bothcontinuous and discrete symmetries of quantum gravity must be gauged. ( − -Form Symmetries from α -Parameters Taking the absence of free parameters in quantum gravity for d > α -parameters. As described in Section 2.1, α -parameters exactlycorrespond to the coupling constants of terms in the Lagrangian for our effective field theory,and so define a ( − − α -parameters violate the condition of no free parameters.Thus, we conclude that the Baby Universe Hypothesis follows from the condition of no freeparameters in quantum gravity.Why do we claim that α -parameters cannot simply be the asymptotic values of dynamicalfields in our effective field theory? Suppose to the contrary that they were, in that thecouplings α i L i arise as the zero modes of interactions φ i L i for some dynamical fields φ i .While it would take infinite energy in order for φ i to differ from (cid:104) φ i (cid:105) throughout all of space,a local variation of φ i must be allowed as a finite energy excitation of the theory by theswampland condition of triviality of cobordism [18]. Thus, within the region where φ i differfrom their asymptotic values, local observers will observe spacetime-dependent α -parameters.If we recall that α -parameters are simply the eigenvalues of the baby universe operators A i ,we immediately see a contradiction. If A i were functions of space, then the baby universescould carry nonzero energy and momentum, which is impossible. Thus, we cannot realizecontinuous α -parameters as arising from dynamical fields, and so the corresponding ( − − d > α -parameters in d > α -parametersexplicitly correspond to superselection sectors in finite space. The condition of triviality For continuous α -parameters, this follows already from taking the fields to have finite mass. ≠ ⟨𝜙⟩ 𝜙 = ⟨𝜙⟩ A finite-energy bubble where the scalar φ differs from its asymptotic value.of cobordism of spacetime [18] exactly says that this is impossible in d >
3, and thatany superselection sectors of quantum gravity must arise as a result of distinct boundaryconditions. It is important to note that while many calculations in [18] are done at the levelof classical configurations, the statement that there cannot be superselection sectors in finitespace is a statement at the level of the full quantum theory. Even if the classical theoryhas no such superselection sectors, it may happen that superselection sectors arise in thequantum theory, just as a quantum theory may have a symmetry that does not act on theclassical configuration space. In fact, the Baby Universe Hypothesis may be viewed as aprecise statement at the quantum level that there can be no superselection sectors in finitespace!
In the previous section, we have explained how previously established swampland conditionsimply the Baby Universe Hypothesis. Conversely, in this section we show that the BabyUniverse Hypothesis implies one of the most basic swampland conditions, that there cannotbe global symmetries in quantum gravity for d >
3. This provides further evidence that theBaby Universe Conjecture is the natural expectation from the perspective of the SwamplandProgram, and indeed it fits neatly in the broader web of swampland conjectures.The argument, in fact, is quite straightforward. Suppose for the sake of contradictionthat there we had a theory of quantum gravity with a global symmetry. As Coleman notes[4], while baby universes may not carry any gauge charge, they can carry charge associatedwith global symmetries, and so the baby universe Hilbert space must break up into the direct12um over sectors with different global charge. These different sectors are all orthogonal evenafter taking the gauge redundancies into account, since the Euclidean path integral cannotviolate conservation of global charge by assumption (otherwise, we wouldn’t have an exactsymmetry). Further, these sectors are nontrivial, since we may act with a nontrivial localoperator carrying the charge to move from one sector to another. Thus, the presence of aglobal symmetry implies that dim H BU > , and so the Baby Universe Hypothesis implies the condition of no global symmetries. In this section, we first review how the Baby Universe Hypothesis is implied by a standardinterpretation of the AdS/CFT dictionary, as well as how it resolves the potential paradoxeswith factorization of correlation functions (both noted in [9]). We also explain why anensemble interpretation of holography in d = 2 or 3 such as JT gravity is actually expectedbased on the standard (non-ensemble) version of holography in higher dimensions, and whywe do not expect an ensemble to play a role for holography in d >
3. Finally, we explainhow the Baby Universe Hypothesis allows us to think of the holographic principle as a formof “Gauss’s law for entropy” in quantum gravity.
Suppose we consider a theory of AdS quantum gravity in ( d + 1) spacetime dimensions. Ifwe fix asymptotic boundary conditions for our bulk theory, in the form of a conformal d -manifold X and boundary conditions J for the bulk fields, we may compute the gravitationalpartition function Z QG ( X, J ) = (cid:90)
X,J D g D φ e − S ( g,φ ) , by integrating over bulk metrics and field configurations with the specified boundary con-ditions. We will supress the argument J for clarity below. While there is no obvious apriori reason for Z QG to depend locally on X, J , the miracle of the standard (non-ensemble)AdS/CFT dictionary is that the gravitational partition function is exactly equal to the par-tition function of a local conformal field theory defined on the boundary, Z QG ( X ) = Z CFT ( X ) .
13s such, Z QG must satisfy the axioms of local quantum field theory. In particular, givena ( d −
1) manifold M , we may define a Hilbert space H QG ( M ) of bulk states which have M as their asymptotic boundary. Furthermore, given two ( d − M , M , we havethat the Hilbert space defined by the disjoint union of M and M must tensor factorize, H QG ( M (cid:116) M ) = H QG ( M ) ⊗ H QG ( M ) . From the bulk perspective, this is also a mystery, since the bulk Hilbert space seems to includeconnected geometries which do not naively factorize. However, motivated by the exampleof the thermofield double state and the two-sided AdS black hole [14], we have learned thatconnected geometries may be realized as entangled superpositions of disconnected geometries.Thus, this factorization also relies on the same gauge redundancies [9, 10] described abovein Section 2.3.What is the Hilbert space H BU of baby universes? By definition, a baby universe stateis a state of closed universes, and as such has no asymptotic boundary. Put differently, wehave that H BU = H QG ( ∅ ) , where ∅ denotes the empty ( d − ∅ is the unit for disjoint union, meaningthat ∅ (cid:116) M = M, for any ( d − M . Together with boundary locality, this implies that we have H QG ( M ) = H BU ⊗ H QG ( M ) . (1)If all the Hilbert spaces involved here were finite-dimensional, this would immediatelyimply dim H BU = 1 , simply by taking the dimension of both sides of (1). Since in general the Hilbert spaces H QG ( M ) will be infinite-dimensional, we need to work a bit harder. One way to argue isin the case that the dual CFT is compact, meaning that the spectrum of the Hamiltonian H on H QG ( M ) is discrete for a compact manifold M . In this case, we can truncate thespectrum at some energy cutoff Λ, and look at the finite-dimensional Hilbert space of stateswith energy E <
Λ. By the Wheeler-de Witt equation, all states in H BU have zero energy,14nd so we have H QG ( M ) (cid:12)(cid:12)(cid:12) E< Λ = H BU ⊗ (cid:16) H QG ( M ) (cid:12)(cid:12)(cid:12) E< Λ (cid:17) . Taking the dimension of both sides again impliesdim H BU = 1 , as desired.A better way to argue in general is to interpret (1) as saying that H BU is a unit object forthe tensor product of Hilbert spaces, which implies that H BU is canonically isomorphic to C . In fact, that quantization on the empty manifold gives a one-dimensional Hilbert spaceis one of the axioms of local quantum field theory, and is completely obvious from the pathintegral, since there is a unique field configuration on the empty manifold. Thus, the BabyUniverse Conjecture follows immediately from a standard interpretation of the holographicdictionary and the axioms of local quantum field theory. One key piece of the standard AdS/CFT dictionary that has received a great deal of recentattention [3] is the factorization of CFT correlation functions. What this means is that ifwe compute the partition function on a disjoint union X (cid:116) X , then the partition functionfactorizes, Z ( X (cid:116) X ) = Z ( X ) Z ( X ) . This is another axiom of local quantum field theory. However, from the perspective of thebulk Euclidean path integral, it is again unclear how this could be the case, as there arecontributions to the disconnected partition function involving connected bulk geometries,and so in order to factorization to arise from a naive bulk path integral there must be enor-mous cancelations between connected and disconnected contributions to the two-boundarypartition function.One way to compute this partition function is to utilize the baby universe Hilbert space,following Marolf and Maxfield [9]. In particular, fixing a boundary manifold X and sources,they define a state |Z ( X ) (cid:105) ∈ H BU , by performing the Euclidean path integral with X as the asymptotic boundary in the past.Alternatively, this state may be defined by acting on the Hartle-Hawking state with the15perator (cid:98) Z ( X ) that inserts an additional asymptotic boundary X , |Z ( X ) (cid:105) = (cid:98) Z ( X ) | HH (cid:105) . Noting that Z ( X ) is nothing but a baby universe operator A i , we see that it must be diagonalin the α -eigenbasis with eigenvalues Z α ( X ), and we may compute |Z ( X ) (cid:105) = (cid:88) α c α Z α ( X ) | α (cid:105) , where c α = (cid:104) α | HH (cid:105) is the amplitude to be in state | α (cid:105) in the Hartle-Hawking wavefunction(which we assume is taken to be normalized).With this machinery in hand, we may immediately compute the partition function, Z ( X ) = (cid:104) HH | (cid:98) Z ( X ) | HH (cid:105) = (cid:88) α p α Z α ( X ) , where p α = | c α | is the probability distribution on α in the Hartle-Hawking ensemble. Fur-ther, we must have p α > α (see [9]), and so this will always be a nontrivial ensembleif there are nontrivial α -parameters. Just as easily, we may compute the two-boundarypartition function, Z ( X (cid:116) X ) = (cid:104) HH | (cid:98) Z ( X ) (cid:98) Z ( X ) | HH (cid:105) = (cid:88) α p α Z α ( X ) Z α ( X ) , and so we see a lack of factorization coming from a nontrivial ensemble of α -parameters. Inorder to have factorization, we could work in a fixed α -eigenstate instead of in the Hartle-Hawking state, where we would find (cid:104) α | (cid:98) Z ( X ) (cid:98) Z ( X ) | α (cid:105) = Z α ( X ) Z α ( X ) = (cid:104) α | (cid:98) Z ( X ) | α (cid:105) (cid:104) α | (cid:98) Z ( X ) | α (cid:105) . This is a manifestation of the fact that fixing a choice of α does indeed define a consistentand local boundary quantum field theory.If the Baby Universe Hypothesis is satisfied, then the Hartle-Hawking state, as the uniquestate in H BU , is in fact itself an α -eigenstate, and so we see that we have factorization nomatter what! Put differently, if the Baby Universe Hypothesis is satisfied, then we have |Z ( X ) (cid:105) = Z ( X ) | HH (cid:105) , X is just proportional to the Hartle-Hawking state, and so factoriza-tion follows immediately from the one-dimensionality of H BU . One consequence of this isthat the Baby Universe Hypothesis requires enormous and miraculous cancelations betweencontributions to the Euclidean path integral with different topologies, a highly non-genericsituation from the perspective of effective field theory. One strong motivation to doubt the amazing cancelations described in the previous section,which are key to the validity of the Baby Universe Conjecture, is the example of JT gravity in d = 2. In JT gravity, the full bulk path integral may be performed without adding any newUV degrees of freedom [3], and indeed, these types of cancelations do not occur. JT gravityis dual [3] to an ensemble average of d = 1 quantum systems given by a random matrixmodel, leading to a consistent new model for holography involving an ensemble. Further, inthis framework, the disconnected partition functions do not factorize, as is expected from anexact quantization of JT gravity [19]. In this section we explain why this is not surprisingin the d = 2 context, and why we should not expect it to generalize to higher dimensions(with the possible exception of d = 3), from yet another perspective.Consider a collection of k quantum systems in d = 1, defined by N × N Hamiltonians H i ( A ) for 1 ≤ i ≤ k that all depend on some background parameters A , leading to k unitarytime evolution operators U i ( A, τ i ) = e − iτ i H i ( A ) , We may then consider the ensemble average over A of their partition function on k copies of S with lengths τ i , given by Z = (cid:90) dA e − S ( A ) k (cid:89) i =1 Tr U i ( A, τ i )We have not yet stated what A , H i ( A ), and S ( A ) are; we now fill this gap. Considera D -dimensional sphere with k circles γ i of lengths τ i on it, and consider an SU ( N ) gaugeconnection A on S D . We identify U i ( A, τ i ) = P exp (cid:18) i (cid:90) γ i A (cid:19) ,
17s the Wilson line observables in the gauge theory. Moreover, we identify S ( A ) = 1 g (cid:90) Tr( F ∧ (cid:63)F ) . We can now reinterpret Z as the expectation value of k Wilson loop observables in a D -dimensional SU ( N ) gauge theory, Z = (cid:42)(cid:89) i Tr U γ i (cid:43) . In other words, from the perspective of the D -dimensional theory, the d = 1 ensemble averageis nothing but the path integral description of the SU ( N ) gauge theory.Now suppose this theory has a large N dual (if necessary we can consider adding addi-tional fields and couplings to the above scenario to realize this). Then we can ask how todescribe the large N dual theory in terms of sum over Riemann surfaces with k boundaries,as suggested by ‘t Hooft [20]. In the context of standard (non-ensemble) version of AdS/CFTin string theory, the corresponding ‘t Hooft surfaces are identified with string worldsheets,which can be viewed as a d = 2 quantum gravitational system. Summing over all Riemannsurfaces with k boundaries is the standard large N description of Z , so we learn that theWilson loops observables of any large N gauge theory which admits a dual description leadsto a d = 2 theory of quantum gravity which computes the average of these d = 1 partitionfunctions over an ensemble! So we seem to have reproduced a scenario analogous to JTgravity in a much more general context. We now explain more explicitly how JT gravity fitsin this picture, which was the motivation for this general construction.Consider SU ( N ) Chern-Simons gauge theory on S . It is known that this theory can berealized by topological A-model strings on T ∗ S with N topological branes wrapping S [21].Using this setup, it has been argued in [22] that this has a large N -dual given by A-modeltopological gravity on the resolved conifold. In this context, the worldsheet diagrams oftopological strings can be viewed as large N ‘t Hooft surfaces. In computing k Wilson loopobservables, we would be considering diagrams with k boundaries (for examples of this see[23], as in the general scenario above). Moreover, the mirror of this picture leads to B-modeltopological string description. In particular, one finds that the large N limit of this matrixmodel is holographically dual to topological gravity on the resolved conifold given by the See [12] for a similar perspective in the context of the c = 1 string. x + y + u + v = 0 . As it was explained in [24] this can be generalized to an arbitrary matrix model whose gravitydual is given by a local Calabi-Yau 3-fold F ( x, y ) + u + v = 0where F ( x, y ) = 0 describes the spectral curve of the matrix model. JT gravity is related[3] to the partition function of Mirzakhani’s model [25] which in turn is given by the matrixmodel with spectral curve [26], F ( x, y ) = y − sin √ x. For an explanation of this based on the relation of Mirzakhani’s model to Mumford classessee [27]. Therefore we see that JT gravity is equivalent to the worldsheet description oftopological gravity on the non-compact Calabi-Yau 3-fold given by y − sin √ x + u + v = 0 . Moreover it is explained in [28] how the Eynard-Orantin rules of computation for large N dual of matrix models [29] can be interpreted as arising from the d = 6 topological gravity ofthe B-model, which is the Kodaira-Spencer theory of gravity in 6 dimensions [30]. Anotherexample of this type is the model studied in [12] of the c = 1 string, which is equivalent atthe self-dual radius to the B-model on the deformed conifold [33].Thus, we have explained why one can think of the average over an ensemble of backgroundparameters for a system of d = 1 as naturally induced from computation of specific correlatorsof a higher dimensional gauge theory, and that the emergence of the d = 2 quantum gravityshould be viewed as a large N dual description ala ‘t Hooft. From this perspective, we can askwhether there can be a higher dimensional generalization to d ≥ The connection of this theory with Mirzakhani’s model can be explained by noting that the (1 , d = 2 topological gravity leads to computation of Mumford classes [31], and thatthis theory before deformation is desribed by the topological B-model on y − x + u + v = 0 [32]. Includingdeformations to convert Mumford classes to Mirzakhani model is equivalent to replacing x with sin √ x asexplained in [27]. It is interesting that Mirzakhani’s model can be viewed as computing the partition functionof a topological string on a non-compact Calabi-Yau 3-fold. d = 2 story to d = 3, where, say, M2 branes maybe viewed as replacing the role of string worldsheets for the theory of large N M5 branes,ending on the surface operators of this theory. Of course there is currently no known wayof thinking of M2 branes in the context of M-theory holography as playing the same rolethat strings play for the large N description of gauge theories. Regardless of whether thiscan be viewed as a d = 3 gravitational system, this perspective clearly suggests that thisexample cannot be generalized to d >
3, reinforcing the arguments we presented based onother swampland principles. It is quite interesting that in this context the fact that d = 2and perhaps d = 3 may be exceptional cases for swampland principles is mirrored by thefact that the only known quantum systems decoupled from gravity involve either particlesor strings. While many discussions of holography occur within the context of AdS/CFT, there is a morebasic holographic principle, due to ‘t Hooft [34] and Susskind [35], that should apply in muchmore generality. In this section, we explain how the Baby Universe Hypothesis naturallyleads us to a picture where we interpret the holographic principle as providing a form of“Gauss’s law for entropy” in quantum gravity.What does Gauss’s law tell us? At its most basic, say in the context of electromagnetism,Gauss’s law comes in two versions, a local form and a global form. The local form tells usthat we may measure the charge inside any region by a local calculation on the boundaryof the region, namely by computing the flux of the electric field through the boundary. Theglobal form tells us that if we take space to be a closed manifold, the net charge must vanish,since electric field lines emanating from a positive charge must end somewhere on an equaland opposite negative charge. Of course, the local and global forms are connected: since theboundary of a closed space is empty, there is no way for it to support a nonzero electric flux.The analogy to the holographic principle is straightforward. In particular, the holographicprinciple tells us that the entropy of a bulk region of space may be computed as the entropyof a state in a local quantum system living on the boundary of the region. This is a localstatement, and should be compared to computing the charge inside a region by the flux ofthe electric field through the boundary. What, then, is the global analog? We claim that20t is precisely the Baby Universe Hypothesis! Indeed, saying that the Hilbert space of babyuniverses is one-dimensional means that any quantum state of closed universes carries noentropy. The arguments in Section 4.1 that identify the baby universe Hilbert space withthe Hilbert space of the dual quantum field theory on the empty manifold are analogous tothe derivation of the global version of Gauss’s law from the local version.It is important to note the key role played by quantum mechanics in this analogy. Inparticular, unlike for classical systems, quantum entanglement allows the entropy of a collec-tion of quantum subsystems to be smaller than the sum of the entropies of each subsystem.In the context of quantum gravity, this may be realized as follows. If we imagine cuttingup a closed universe into many regions with boundaries, each boundary can have nonzeroarea, and so each region can have nonzero entropy. However, in order to glue the subregionstogether into a closed universe, we must choose specific entangled states on the boundaries(corresponding to sewing the geometries together), and in this way end up with a much lowerentropy (namely, zero) than the sum of the entropies of each region, as required by the BabyUniverse Hypothesis.
In this note, we argued that the Baby Universe Hypothesis, which was noted in [9] as onepotential resolution to the paradoxes of the Euclidean path integral, is in fact the naturalresolution from the perspective of the Swampland Program. Further, it provides a cleansynthesis of many things we hope to be true about quantum gravity, including the validityof the Euclidean path integral, the absence of free parameters in d >
3, and the standardunderstanding of the AdS/CFT dictionary. We interpreted the possibility of an ensembleaverage in d = 2 and potentially 3 as arising naturally from worldvolume perturbativeexpansions of a larger theory of quantum gravity that does indeed satisfy the Baby UniverseHypothesis. Thus, while we cannot prove the truth of the Baby Universe Hypothesis (sincewe do not have a complete theory of quantum gravity with d > cknowledgments We would like to thank Robbert Dijkgraaf, Dan Freed, Arthur Hebecker, Simeon Hellerman,Daniel Jafferis, Juan Maldacena, Donald Marolf, Henry Maxfield, Miguel Montero, GeorgesObeid, Steve Shenker, Pablo Soler, and Irene Valenzuela for useful discussions. We havegreatly benefited from the hospitality of UC Santa Barbara KITP where this project wascompleted.The research of C.V. is supported in part by the NSF grant PHY-1719924 and by a grantfrom the Simons Foundation (602883, CV). This research was supported in part by the Na-tional Science Foundation under Grant No. NSF PHY-1748958. This material is based uponwork supported by the National Science Foundation Graduate Research Fellowship Programunder Grant No. DGE1745303. Any opinions, findings, and conclusions or recommendationsexpressed in this material are those of the authors and do not necessarily reflect the viewsof the National Science Foundation.
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